# Usage of 'Less than' condition in Sum

I have following sum: $$\sum_{\substack{n,j\\j. How can I give this $$j condition in Sum?

• Table[x[n] x[j], {n, 1, 3}, {j, n, 3}] // Flatten // Total – cvgmt Nov 12 '20 at 14:08
• Sorry @cvgmt, j shall be lower n. This is the right: Table[x[n] x[j], {n, 1, 3}, {j, 1, n - 1}] // Flatten // Total  – Akku14 Nov 12 '20 at 14:30
• @Akku14 Thanks you, you are right. – cvgmt Nov 12 '20 at 14:33

For more complex condition:

Sum[Boole[j<n] x[n] x[j] ,{n,1,3},{j,1,3}]


For this simple condition:

@cvgmt give it as comment.

Just for fun:

Subsets[Array[x, 3], {2}] // Map[Apply@Times] // Total

• thank you. I have used Boole to provide equal or not equal condition in sum previously; I was confused about whether it can be useful in this case also, – Arghya Datta Nov 12 '20 at 17:18

The most direct solution is simply:

Sum[x[n] x[j], {n, 1, 3}, {j, 1, n}]


(* x[1]^2 + x[1]*x[2] + x[2]^2 + x[1]*x[3] + x[2]*x[3] + x[3]^2 *)

The same sort of thing works with NSum, Product and Product.

• I think you are almost right. Just the upper limit of j should be n-1. – Arghya Datta Nov 12 '20 at 16:49
SparseArray[{n_, j_} /; j < n -> Subscript[x, n] Subscript[x, j], {3,
3}] // Flatten // Total


As usual, there are several ways to do it with Mathematica depending on the specific features that you want. Suppose you want to do some processing on all cases where $$i where $$i,j$$ are in some range or list. The following code, in order of complexity, will each produce such a list of tuples:

r = Range[3];
Subsets[r, {2}]
Cases[Tuples[r, 2],  {i_,j_}/;i<j]
Select[Tuples[r, 2], #[[1]]<#[[2]]&]
Flatten[Table[If[i<j, {i,j}, Nothing], {i,r},{j,r}]


Once you have such a list L of {i,j} tuples, then you can find the sum of f[i,j] over all tuples with code such as:

Plus@@(L /. {i_,j_}->f[i,j])
Plus@@(# /. ij_->f@@ij& /@ L)


Same thing with the product using Times instead of Sum.